Suppose that two students from your school have just been
apprehended by the police for allegedly possessing illicit
substances, possibly with the intention of reselling (or
"trafficking in") these substances. For identification
purposes, lets refer to the first student as Psmith. The
second student will be known as Joans.

The police place Psmith and Joans in separate cells, not
allowing them to communicate with each other, and interrogate
them in separate interview rooms. During the interrogations, each
prisoner is offered the following choice: if you confess to being
a dealer in partnership with the other prisoner, you will go free
and the other prisoner will be imprisoned for ten years. Of
course, though, you cannot both be witnesses for the Crown; if
you both confess, you will each receive prison sentences of six
years. But if neither of you confesses to being a dealer, we will
still be able to convict you of possession, and you will be
imprisoned for one year.

These options generate the dilemma for Psmith and Joans. While
each one would like to go free and avoid even one year in prison,
they both have some misgivings about finking on their compatriot.
At the same time, they have little or no desire to be the patsy
in the case and serve the full ten years while the other goes
free. If they could coordinate their behaviour, they would each
likely remain silent and serve just one year. In the absence of
coordination, though, they are both just as likely to confess and
end up serving six-year sentences. The source of their dilemma,
then, is their inability to communicate and coordinate their
actions.

The choices available to Psmith and Joans, along with the
possible outcomes, are shown in Figure 12-1, which is called a payoff
matrix. Psmiths two choices are listed down the left
side of the matrix: confess and dont confess. Joans
two choices are listed across the top of the matrix: again,
confess and dont confess.

There are four possible outcomes of the choices the two
students might make. These outcomes are outlined with dark lines;
each possible outcome is called a cell in the payoff
matrix.

Joans

Choices

Dont Confess

Confess

Dont

1 year

no time

Psmiths

Confess

1 year

10
years

Choices

Confess

10 years

6 years

no
time

6
years

Figure 12-1 The
Prisoners Dilemma Payoff Matrix

Within each cell is a diagonal line (in the text, not here on
the website). The number above and to the right of the diagonal
line shows the sentence that Joans will receive if the two of
them choose that combination of strategies; Psmiths
sentence is shown below and to the left of the diagonal line. For
example, if Joans were to remain silent but Psmith confessed, the
outcome would be shown in the lower left-hand cell of the payoff
matrix  Joans would receive a ten-year sentence, and Psmith
would go free (i.e. receive a zero sentence  do no time).

Whether Psmith or Joans (or both or neither) will confess
cannot be answered by relying only on the payoff matrix and
whatever underlying mathematics one may wish to invoke to study
the problem. The optimal choice for each of them can emerge only
after we know something else about the situation. The additional
information necessary to solve the dilemma falls into three broad
categories: (1) the expected detection lag, (2) the expected
retaliation lag, and (3) the expected size of the retaliation.
Lets examine each of these in turn.

Expected
Detection Lag

If Psmith (for some unfathomable reason)
believes that Joans will never find out that Psmith confessed,
then Psmith might give very serious consideration to confessing.
"After all", the reasoning might go, "If I
confess, then I can go free and Joans will never figure out what
happened." If, at the same time, Joans goes through the same
inexplicable thought process, Joans will also see that there are
large expected benefits from confessing and zero expected costs
from confessing. And if they both reason through the problem in
this way, they will both confess; and they will both end up
serving six years. You might well wonder how either Psmith or
Joans could be so stupid as to imagine the other would not figure
out they had confessed. You are probably correct. But this
scenario has considerable applicability later in the chapter.

Suppose that Psmith, knowing that he has
an incentive to confess, realizes that Joans also has an
incentive to confess due to the long (infinite, with our
assumptions) detection lag. In this case even if Psmith, for some
reason such as honour or loyalty, had wanted to remain silent, he
will grow quite concerned about what Joans will do. If Psmith
anticipates that with high probability Joans will confess, then
Psmith can make himself better off by confessing. A six-year
sentence is clearly better than the ten-year sentence he would
receive if Joans confesses and he doesnt. Joans will likely
go through the same type of thought process: "If Psmith is
going to confess, Ill be better off if I do, too, so I
might as well." Even if Psmith and Joans are wrong in their
expectations, so long as they believe that the detection
lag is long, then they have little or nothing to fear in the way
of retaliation from the other, and they will be likely to
confess. But if the detection lag is short, or if the expected
probability of detection is high, then whether they decided to
confess will depend on what they expect the consequences will be
 what will happen to them if they confess and the other
finds out they confessed.

Expected
Retaliation Lag

The consequences that Psmith and Joans
will anticipate if they confess have two components: time and
size. The time component is called the retaliation lag. If
Psmith believes that Joans will not be able to retaliate for a
long time, then Psmith might be more likely to confess. For
example, if Psmith thinks there is a chance Joans will not
confess, Psmith can confess and anticipate being free from
retaliation by Joans for ten years. Even when the detection lag
is short, if the retaliation lag is this long, Psmith might
consider confessing.

Of course, if Psmith considers
confessing under these conditions, he must expect that Joans will
do so, too. And expecting that Joans is likely to confess, Psmith
will weigh the options of six years versus ten years and be even
more likely to confess.

But just because Joans is locked away
for six or ten years doesnt mean that the retaliation lag
would necessarily be that long. If Psmith breaks some "code
of honour" by confessing, Joans may well have some friends
or family members who could inflict some form of retaliation on
Psmith or on his friends and family within a very short period of
time. If, for example, Psmith and Joans are members of an
organization that shares this code of honour, they might very
well fear that retaliation would be swift. And if they have good
reason to fear swift retaliation, they will probably think twice
about confessing. The effect is that both Psmith and Joans will
be better off, on average, if they believe that the other person
would have both a short detection lag and a short retaliation
lag; under these circumstances, neither would be as likely to
confess, and they would each serve only one year in prison.

Expected
Size of the Retaliation

Even if Joans can detect that Psmith has
confessed, and even if Joans is able to retaliate quickly, Psmith
may still choose to confess if he believes that Joans will not be
able to inflict much punishment on him for his having confessed.
If the worst that Joans can do in the way of retaliation is to
say, "Well, the next time Im going to confess,"
or "Im not going to be your friend anymore," then
Psmith may decide that this is not going to be very much
deterrence and confess anyway. And if the possibilities are
symmetrical, i.e. if Psmith can similarly impose no harsh
penalties on Joans for confessing, then they will both have an
incentive to confess, and they will both end up serving six
years. Furthermore, each student, knowing that the other expects
little in the way of deterrence, will not want to be left holding
the bag, and will decide to confess as well. Witness protection
programs are designed to reduce the size of the expected
retaliation by making it extremely unlikely that Joans could
inflict suffering on Psmith should Psmith decide to confess.

If, however, Psmith fears that Joans (or
some of Joans associates) will inflict serious punishment
on Psmith or possibly some members of Psmiths family,
should he confess, then even if the retaliation lag is fairly
long, Psmith may have serious qualms about whether to confess.
The phrase, "Ill hunt you down, even if it takes the
rest of my life," conveys this threat quite effectively. And
if both Psmith and Joans believe the other will carry out
substantial retaliation, they will both choose not confessing
rather than confessing.

Notice that if the students are similar
in their abilities and their expectations, then the likely
outcome of the game is that either both will confess or neither
will confess. If one of them thinks it advantageous to confess,
it is quite likely that the other one will, too. And if they
think the other one will confess, they will realize that their
own best strategy will be to confess  sort of making the
best of a bad situation. By the same token, if one of them thinks
the other is likely to be able to detect the confession and
retaliate swiftly and strongly, they will be unlikely to confess
 and by symmetry they are both likely to assess the
situation this way. The conclusion, then, is that the most likely
outcome is that both will choose to do the same thing, either
confess or not confess; it is unlikely that one will choose to
confess while the other chooses to remain silent. And whether
they both choose to confess will depend on the expected length of
the detection lag, the expected length of the retaliation lag,
and the expected size of the retaliation.

LAGS AND
PROBABILITIES... (see the text)

INTERDEPENDENCE
AND BUSINESS DECISIONS

Now lets see how the
prisoners dilemma game can help us understand business
decisions when rivals are aware of their interdependencies.
Suppose that Smith and Jones are direct rivals in business,
selling very similar products to the same potential pool of
customers. Up until now, Smith and Jones have been cooperating,
and charging a price that would maximize their joint returns,
each earning 14% on their invested capital. But because there are
laws against explicit price coordination, as we shall see in the
next chapter, Smith and Jones are not allowed to continue their
explicit cooperation. What will they do?

To answer this question, we begin by
setting out the options, and to keep the analysis simple, we will
limit their choices to just two options: cut prices or keep
prices at the current level. Smith considers cutting her price.
She anticipates that if she does so, and if Jones doesnt
follow suit, she can earn a 20% rate of return, while Jones will
earn only 8%. However, if Jones matches her price cut, they will
each earn a 10% rate of return. These options, along with the
expected payoffs to both players, are shown in the payoff matrix
in Figure 12-2.

Jones

Choices

Dont Cut Prices

Cut Prices

Dont Cut

14%

20%

Smiths

Prices

14%

8%

Choices

Cut

8%

10%

Prices

20%

10%

Figure 12-2 A
Price-Cutting Payoff Matrix

The problems of the prisoners dilemma appear again: both
players realize that they can be better off if they cut prices
while the other one doesnt, but they realize they will be
worse off if they both cut prices. Should they cut prices or keep
them at their current levels?

DETECTION
LAG (AGAIN)

If Smith can cut prices without Jones
finding out about the price cut, Smith can move to the lower
left-hand cell of the payoff matrix and earn a 20% rate of
return, leaving Jones with only an 8% rate of return. Jones would
have to be pretty stupid not to see the price change in this
situation, though. All it would take would be a visit to
Smiths store to check prices now and then. But Jones could
probably figure it out even without the visits: if Jones
sales begin falling precipitously for no apparent reason, Jones
might well begin to suspect that Smith has cut her prices. In
fact some customers, trying to get a better price from Jones
might say, "Ill buy from you if you can match
Smiths lower prices; otherwise Ill go to Smith."

To counter this possibility, firms
sometimes do not announce their price cuts. By not issuing new
price lists, the firms can then offer different discounts to
different customers  price discrimination, as we saw in
Chapter 11. But they also increase the detection lag. By offering
secret discounts to some customers, and pleading with them not to
tell Jones, Smith can increase her profits and delay the time
when Jones becomes aware of her price cuts.

The customers may at first wonder why
they should go along with Smith. After all, they might be able to
get even lower prices from Jones by telling him about
Smiths new, lower prices. The only reason they will go
along with Smith would be that they are involved in a game with
Smith, too. If they tell Jones about Smiths lower prices,
they might be able to get an even lower price from Jones now. But
then Smith (if she finds out they told Jones) might not offer the
secret price cuts in the future. If they are frequent customers
of Smith, this will be important to them over the longer run, and
they will be more likely to honour Smiths request. If,
however, they purchase something from Smith only rarely, and
especially if the item is high-priced, they might well be better
off by violating Smiths confidence, telling Jones, and
trying to get an even lower price from Jones. Even if Smith tries
to retaliate by not offering a price cut in the future, if the
retaliation lag is long and the expected gains to the customer
from telling Jones are large, customers will be likely to tell
Jones. And knowing this, Smith will be less likely to try to
offer secret discounts to these customers.

Even if Smith is able to grant off-list
discounts to her customers, though, the chances are good that
Jones will eventually learn about her price cuts (by noticing
that his own sales are declining, if nothing else), and Jones is
not likely to sit on his thumbs while Smith rakes in the extra
profits.

RETALIATION
LAG (AGAIN)

How long will it take Jones to cut his
prices to match Smiths? With computerized pricing, even in
large bureaucracies Jones can key in a few numbers and have
prices changed in all his branches within hours, if not minutes.
The retaliation lag in this case is short. But if it takes some
time to prepare new advertisements with the lower prices and to
place the ads strategically, the retaliation lag might be as much
as several weeks or even a month or two. Meanwhile Smith has a
price advantage and is cleaning up.

Different marketing tactics have
different expected retaliatory lags. Price cuts are often quite
easy to match fairly quickly; they have a short detection lag and
a short retaliatory lag. But unique ad campaigns, while easy to
detect quickly, are more difficult to counter quickly. So when
there are comparatively few rivals squaring off in head-to-head
competition, we often see them engaging in what we call non-price
competition. Price competition is much rarer, though not
completely absent.

EXPECTED
SIZE OF THE RETALIATION (AGAIN)

What is the worst that Jones can do to
retaliate against Smiths price cuts? What type of deterrent
action can Jones take that will serve as an effective threat
against further price cuts by Smith? Ruling out illegal and
violent action, about the only options available to Jones are for
him to change his own marketing program. Changing the price to
match a price cut will often be effective in sending a signal to
Smith  "cut prices, and Ill have to match you
even though well both be worse off. At least Ill be
less bad off." Sometimes, if Jones fears that Smith might
then cut prices again, Jones might want to make a pre-emptive
move by cutting prices significantly at the outset, to send a
strong signal to Smith not to cut prices again.

Changing his product design or changing
an advertising campaign might or might not be a large
retaliation. The effects of these types of changes often are
highly uncertain; some work out very successfully, while others
are dismal failures. Even though the businesses would not intentionally
choose an unsuccessful marketing strategy, many plans do not work
out as hoped.

Applying the conclusions from game
theory to the business world, we can see that for choices
involving long detection lags, long retaliation lags, and small
retaliatory measures, rivals are likely not to cooperate with
each other, and they will end up not maximizing their joint
returns. The cooperation need not involve face-to-face
conversations and meetings, although increased communications are
sure to reduce the length of detection lags and to give the game
players a better idea of the likelihood of different reactions
from their rivals. But cooperation, even if it is tacit, will be
more likely when the detection and retaliation lags are short or
if the expected size of the retaliation is large.

Suppose that instead of cutting prices,
Smith is considering introducing a new product line. We can use
the prisoners dilemma to see why this might be the option
many rivals would choose instead of price-cutting to increase
their profits. If Smith can do the research and development on
the product in secrecy and then launch the product with a giant
marketing blitz, Jones will be left in the dust. Smith can earn
big profits, and Jones will not be able to counter Smiths
moves for quite some time. The detection lag is long, when you
think of how long it takes Jones to find out that Smith is
planning the move. And the retaliation lag is long, too, once
Jones finds out what Smith is up to. Jones must develop his own
new product line and marketing campaign, and these things cannot
ordinarily be done quickly.

Knowing that Jones will not be able to
counter her moves very quickly, Smith gives serious consideration
to this tactic. But at the same time, she knows that Jones is
thinking the same thing. She never knows when Jones will launch a
new product, and she doesnt want to be left biding her time
while Jones new product cuts into her profits.
Consequently, even if Jones were not planning to develop a
new product, Smith will anticipate that Jones might be
working on one, and so she will continue new product research and
development as a defensive strategy, if not as an offensive
strategy.

The game theoretic approach seems to
imply that rivals in these types of situations will not cut
prices very often but will always be working on new ad campaigns
or the development of new product lines. These implications are
not always borne out, however. For some products, prices are
changed frequently, and there are comparatively few new
advertising campaigns. Nevertheless, game theory has a general
applicability that helps us analyze peoples choices in many
different situations. We will use it repeatedly in the next few
chapters as we examine the roles of different people and
institutions in the economy.

... continued in the textbook...

(sample) QUESTIONS
FOR DISCUSSION

Why are there gasoline price wars? Before you answer this
question, what would the prisoners dilemma imply
about price wars? If one dealer or supplier or refiner
cuts the price, do you think they dont anticipate
that others in the industry will retaliate and match the
lower prices? If they think their price cuts will be
matched, why do they bother starting the price war in the
first place?

Do most drivers buy more gas when there is a price war?
Do they end up "filling the top half instead of the
bottom half" of their gas tanks? What happens to the
amount of gasoline stored in customers gas tanks?
If it is extremely costly for oil companies build more
storage facilities, and if it is illegal to dump
petroleum products into the ground, might an oil company
be willing to start a price war just to encourage
customers to store more gasoline?

Experiments involving students playing forms of the
prisoners dilemma game have found that the players
are able to discover and play a joint maximizing strategy
very quickly. The experiments typically run as follows:
the players are told that they are playing with one other
player who is in a different room. If they both push a
blue button, they will each earn one dollar, but if they
both push the red button, they will receive only 25¢
each. If one pushes the blue button while the other
pushes the red button, the one pushing the red button
will earn two dollars, but the one pushing the blue
button will earn nothing.

Complete the payoff matrix for this game in Figure 12-4.

Player

As Choices

Blue Button

Red
Button

Player Bs

Choices

Blue
Button

Red
Button

Figure 12-4. Experimental Payoff
Matrix

What do you think happens when the players are told,
"This is the last play of the game."?

If you expect that type of behaviour on the last play of
the game, what do you think happens on the 19th
play of game when the players know there will be only 20
plays in the game? The 18th? Why do players
cooperate at all when they play games like this?

There is some evidence that students from arts and
humanities cooperate more and hence have greater total
winnings, playing these games, than do students in
economics and business. Why might this be?